# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a340692 Showing 1-1 of 1 %I A340692 #12 Apr 09 2021 09:41:08 %S A340692 0,0,2,0,4,2,8,4,14,12,26,22,44,44,76,78,126,138,206,228,330,378,524, %T A340692 602,814,950,1252,1466,1900,2238,2854,3362,4236,5006,6232,7356,9078, %U A340692 10720,13118,15470,18800,22152,26744,31456,37772,44368,53002,62134,73894 %N A340692 Number of integer partitions of n of odd rank. %C A340692 The Dyson rank of a nonempty partition is its maximum part minus its length. The rank of an empty partition is undefined. %H A340692 Freeman J. Dyson, A new symmetry of partitions, Journal of Combinatorial Theory 7.1 (1969): 56-61. %H A340692 FindStat, St000145: The Dyson rank of a partition %F A340692 Having odd rank is preserved under conjugation, and self-conjugate partitions cannot have odd rank, so a(n) = 2*A101707(n) for n > 0. %e A340692 The a(0) = 0 through a(9) = 12 partitions (empty columns indicated by dots): %e A340692 . . (2) . (4) (32) (6) (52) (8) (54) %e A340692 (11) (31) (221) (33) (421) (53) (72) %e A340692 (211) (51) (3211) (71) (432) %e A340692 (1111) (222) (22111) (422) (441) %e A340692 (411) (431) (621) %e A340692 (3111) (611) (3222) %e A340692 (21111) (3221) (3321) %e A340692 (111111) (3311) (5211) %e A340692 (5111) (22221) %e A340692 (22211) (42111) %e A340692 (41111) (321111) %e A340692 (311111) (2211111) %e A340692 (2111111) %e A340692 (11111111) %t A340692 Table[Length[Select[IntegerPartitions[n],OddQ[Max[#]-Length[#]]&]],{n,0,30}] %Y A340692 Note: A-numbers of Heinz-number sequences are in parentheses below. %Y A340692 The case of length/maximum instead of rank is A027193 (A026424/A244991). %Y A340692 The case of odd positive rank is A101707 is (A340604). %Y A340692 The strict case is A117193. %Y A340692 The even version is A340601 (A340602). %Y A340692 The Heinz numbers of these partitions are (A340603). %Y A340692 A072233 counts partitions by sum and length. %Y A340692 A168659 counts partitions whose length is divisible by maximum. %Y A340692 A200750 counts partitions whose length and maximum are relatively prime. %Y A340692 - Rank - %Y A340692 A047993 counts partitions of rank 0 (A106529). %Y A340692 A063995/A105806 count partitions by Dyson rank. %Y A340692 A064173 counts partitions of positive/negative rank (A340787/A340788). %Y A340692 A064174 counts partitions of nonpositive/nonnegative rank (A324521/A324562). %Y A340692 A101198 counts partitions of rank 1 (A325233). %Y A340692 A101708 counts partitions of even positive rank (A340605). %Y A340692 A257541 gives the rank of the partition with Heinz number n. %Y A340692 A324520 counts partitions with rank equal to least part (A324519). %Y A340692 - Odd - %Y A340692 A000009 counts partitions into odd parts (A066208). %Y A340692 A026804 counts partitions whose least part is odd. %Y A340692 A058695 counts partitions of odd numbers (A300063). %Y A340692 A067659 counts strict partitions of odd length (A030059). %Y A340692 A160786 counts odd-length partitions of odd numbers (A300272). %Y A340692 A339890 counts factorizations of odd length. %Y A340692 A340385 counts partitions of odd length and maximum (A340386). %Y A340692 Cf. A003114, A006141, A027187, A039900, A067538, A096401, A117409, A143773, A324518, A325134, A340828, A340854/A340855. %K A340692 nonn %O A340692 0,3 %A A340692 _Gus Wiseman_, Jan 29 2021 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE